In Ref. [1]} the interaction becomes smaller compared to Hubble after matter-radiation equality, as \(H\propto a^{-3/2}\) 
while the interaction still drops as \(\Gamma \propto a^{-2}\) , thus the shut-off time is at the time of equality \(\eta _{eq.}\) 
In contrast, for the WZDR+ model the momentum transfer rate shuts off once the temperature drops below \(m_\phi \)  (see Eq. (REF )). This is shown in the bottom panel of Figure REF .
As a result the matter power spectrum, through the \(S_8\)  measurements, is sensitive to \(m_\phi \)  or equivalently to \(z_t\) .
